INTRODUCTION TO TRANSMISSION LINES PART II

DR. FARID FARAHMAND FALL 2012 Transmission Line Model Perfect Conductor and Perfect Dielectric (notes) Simulation Example Transmission Line Model Transmission-Line Equations

Remember:

Kirchhoff Voltage Law:

Vin-Vout – VR’ – VL’=0

Kirchhoff Current Law:

jθ Ae = Acos(θ ) + Aj sin(θ ) Iin – Iout – Ic’ – IG’=0 cos(θ ) = ARe[Ae jθ ] Note: sin( ) AIm[Ae jθ ] θ = E(z) =| E(z) | e jθz VL=L . di/dt Ic=C . dv/dt | e jθ | 1 = B C = A + jB → θ = tan ;| C |= A2 + B2 A

Transmission-Line Equations

ac signals: use phasors

Transmission Line Equation in Phasor Form Derivation of Wave Equations

Transmission Line Equation First Order Coupled Equations! WE WANT UNCOUPLED FORM! attenuation constant complex propagation (Neper/m) constant Phase constant Combining the two equations leads to:

Second-order differential equation Wave Equations for Transmission Line

Impedance and Shunt Admittance of the line Pay Attention to UNITS! Solution of Wave Equations (cont.)

Characteristic Impedance of the Line (ohm)

Using: Proposed form of solution: It follows that:

So What does V+ and V- Represent?

Pay att. To Direction Solution of Wave Equations (cont.)

So, V(z) and I(z) have two parts:

Applet for standing wave: http://www.physics.smu.edu/~olness/www/05fall1320/applet/pipe-waves.html Perfect Example: Air-Line Conductorà Rs=0àR’ = 0 Perfect Dielec Assume the following waves: à COND=0 à 6 G’=0 V (z,t) = 10 cos(2π ⋅700⋅10 − 20z + 5) I(z,t) = 0.2cos(2π ⋅700⋅106 − 20z + 5) Assume having perfect dielectric insulator and the wire have perfect conductivity with no loss Draw the transmission line model and Find C’ and L’; Assume perfect conductor and perfect dielectric materials are used!

Note: If atten. Is zero àreal part MUST be zero! Transmission Line Characteristics

¨ Line characterization ¤ Propagation Constant (function of frequency) ¤ Impedance (function of frequency) n Lossy or Losless

¨ If lossless (low ohmic losses) ¤ Very high conductivity for the insulator ¤ Negligible conductivity for the dielectric Lossless Transmission Line

If

Then:

Non-dispersive line: All frequency components have the same speed! What is Zo? The Big Idea…. Impedance is measured at difference points in the circuit!

ZL V+o Zo Zin

What is the voltage/current magnitude at different points on the line in the presence of load?? Voltage Reflection Coefficient Consider looking from the Load point of view

At the load (z = 0):

Reflection coefficient

The smaller the better!

Normalized load impedance Expressing wave in phasor form:

¨ Remember:

¨ If lossless ¤ no attenuation constant

All of these wave representations are along the Transmission Line Special Line Conditions (Lossless)

¨ Matching line

¤ ZL=Zo àΓ=0; Vref=0 ¨ Open Circuit

¤ ZL=INF àΓ=1; Vref=Vinc ¨ Short Circuit Remember: Everything is with respect ¤ ZL=0 àΓ=-1; Vref=-Vinc to the load so far!

Vref= Vreflected ; Vinc = Vincident Voltage Reflection Coefficient

Pay attention!

Normalized load impedance Example

Example

Example

Notes We are interested to know what Standing Waves happens to the magnitude of the | V| as such interference is created! Finding Voltage Magnitude

When lossless!

Remember: Standing wave is created due to interference between the traveling waves (incident & reflected)

Note: When there is no REFLECTION Coef. Of Ref. = 0 à No standing wave! Standing Wave http://www.falstad.com/circuit/e-tlstand.html

Due to standing wave the received wave at the load is now different Standing Waves Finding Voltage Magnitude

voltage magnitude at z= -d

current magnitude at the source

Let’s see how the magnitude looks like at different z Remember max current occurs values! where minimum voltage occurs (indicating the two waves are interfering destructively)! Standing Wave Patterns for 3 Types of Loads (Matched, Open, Short)

No reflection, No standing wave

¨ Matching line

¤ ZL=Zo àΓ=0; Vref=0

¨ Short Circuit

¤ ZL=0 àΓ=-1; Vref=-Vinc (angle –/+π)

¨ Open Circuit

¤ ZL=INF àΓ=1; Vref=Vinc (angle is 0)

Remember max current occurs where minimum voltage occurs! Standing Wave Patterns for 3 Types of Loads (Matched, Open, Short)

No reflection, No standing wave

BUT¨ Matching WHEN line DO

¤ Z =Z à =0; Vref=0 MAXL o &Γ MIN ¨ Short Circuit

¤ Z =0 à =-1; Vref=-Vinc (angle –/+ ) VoltagesL Γ Occur? π ¨ Open Circuit

¤ ZL=INF àΓ=1; Vref=Vinc (angle is 0)

Remember max current occurs where minimum voltage occurs! Finding Maxima & Minima Of Voltage Magnitude

S = Voltage Standing Wave Ratio (VSWR)

For a matched load: S = 1

For a short, open, or purely reactive load: S(open)=S(short) = INF where |Γ|=1;

Example Measuring ZL with a Slotted Line

Slotted Coaxial Line What is the Reflection Coefficient (Γd) at any point away from the load? (assume lossless line)

At a distance d from the load:

Wave impedance Example http://www.bessernet.com/Ereflecto/tutorialFrameset.htm http://www.amanogawa.com/archive/Trans1/Trans1-2.html

Example

Notes Input Impedance

Wave Impedance Zd

At input, d = l: Short-Circuit/Open-Circuit Method

¨ For a line of known length l, measurements of its input impedance, one when terminated in a short and another when terminated in an open, can be

used to find its characteristic impedance Z0 and electrical length Standing Wave Properties Power Flow

¨ How much power is flowing and reflected? ¤ Instantaneous P(d,t) = v(d,t).i(d,t) n Incident n Reflected i r ¤ Average power: Pav = Pav + Pav n Time-domain Approach n Phasor-domain Approach (z and t independent) n ½ Re{I*(z) . V(z)} Average Power (Phasor Approach)

Avg Power: ½ Re{I(z) * V_(z)}

Fraction of power reflected! Summary Practice 1- Assume the load is 100 + j50 connected to a 50 ohm line. Find coefficient of reflection (mag, & angle) and SWR. Is it matched well? 2- For a 50 ohm lossless transmission line terminated in a load impedance ZL=100 + j50 ohm, determine the fraction of the average incident power reflected by the load. Also, what is the magnitude of the average reflected power if |Vo|=1? 3- Make sure you understand the slotted line problem. 4- Complete the Simulation Lab answer the following questions: - Remove the MLOC so the TEE will be open. How does the result change? Take a snapshot. Briefly explain. - In the original circuit, what happen if we use paper as the dielectric (paper has er of 3.85). Take a snapshot. Briefly explain. - For the obtained Zo in your Smith Chart calculate the admittance. You must show all your work. - What exactly is mag(S11)? How is it different from coefficient of reflection? Is the reflection of coefficient measured at the source or load? - What happens if the impedance of the source (TERM1) is changed to 25 ohm? How does the impedance on the smith chart change? - How do you calculate the effective length?